Interaction between light and matter is one of the most basic processes in nature and represents a cornerstone in our understanding of a broad range of physical phenom- ena. The increasing level of control over light-matter interactions with atomic and solid-state systems has brought forth a new class of quantum many-body systems realized on photon lattices where light and matter play equally important roles in emergent phenomena. Systems of strongly interacting atoms and photons, that can be realized by wiring up individual cavity QED systems into lattices, are perceived as a new platform for engineering synthetic hybrid light-matter systems. While sharing important properties with other systems of interacting quantum particles, here we argue that the nature of light-matter interaction gives rise to unique features with no analogs in condensed matter or atomic physics setups. The central topic of the thesis is a lattice of cavity QED systems described by the Rabi-Hubbard model. We de- scribe the most prominent features of the model associated with quantum criticality. We consider a realistic case of the system open to the environment and investigate thermal radiation from a lattice of cavity QED systems. Next, we demonstrate that the output radiation displays unique features associated with collective excitations of light and matter. Further, we consider a non-equilibrium lattice of cavity QED systems and demonstrate exotic attractors in the phase diagram, associated with the action of the environment, not present in the equilibrium analogs. We conclude the discussions with a theory of measurement applied to the non-equilibrium Dicke model and compare our findings to a recent experiment.